54.4.55 problem 1513

Internal problem ID [12777]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1513
Date solved : Friday, October 03, 2025 at 03:47:23 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }-4 x^{2} y^{\prime \prime }+\left (x^{2}+8\right ) x y^{\prime }-2 \left (x^{2}+4\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 18

ode:=x^3*diff(diff(diff(y(x),x),x),x)-4*x^2*diff(diff(y(x),x),x)+(x^2+8)*x*diff(y(x),x)-2*(x^2+4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (\cos \left (x \right ) c_2 +\sin \left (x \right ) c_3 +c_1 x \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 40

ode=-2*(4 + x^2)*y[x] + x*(8 + x^2)*D[y[x],x] - 4*x^2*D[y[x],{x,2}] + x^3*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^2 \left (c_2 \int _1^xj_1(K[1])dK[1]+c_3 \int _1^xy_1(K[2])dK[2]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 3)) - 4*x**2*Derivative(y(x), (x, 2)) + x*(x**2 + 8)*Derivative(y(x), x) - (2*x**2 + 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3*Derivative(y(x), (x, 3)) + 2*x**2*y(x) + 4*x**2*Derivative(y(x), (x, 2)) + 8*y(x))/(x*(x**2 + 8)) cannot be solved by the factorable group method